Posted by **Anna** on Tuesday, December 7, 2010 at 8:40pm.

If the volume of a cube is increasing at 24 in^3/min and the surface area of the cube is increasing at 12 in^2/min, what is the length of each edge of the cube?

I know that dV/dt=24 and ds/ds=12. I also know that Volume=s^3 and Surface Area=6*s^2, but what do I do from there?

- Calculus -
**Reiny**, Tuesday, December 7, 2010 at 9:22pm
from V = x^3

dV/dt = 3x^2 dx/dt

dx/dt = 24/(3x^2) = 8/x^2

from A = 6x^2

dA/dt = 12x dx/dt

dx/dt = 12/(12x) = 1/x

then 8/x^2 = 1/x

x^2 = 8x

x = 8

- Calculus -
**MathMate**, Tuesday, December 7, 2010 at 9:26pm
Let the independent variable be

s = length of one side of the cube.

As you mentioned,

V(s) = s³

S(s) = 6s²

dV/dt = dV/ds*ds/dt = 3s² ds/dt = 24 ... (1)

dS/ds = dS/ds*ds/dt = 12s ds/dt = 12 ...(2)

Solve for s and ds/dt by substitution or dividing (1) by (2).

- Calculus -
**MathMate**, Tuesday, December 7, 2010 at 9:29pm
Let the independent variable be

s = length of one side of the cube.

As you mentioned,

V(s) = s³

S(s) = 6s²

dV/dt = dV/ds*ds/dt = 3s² ds/dt = 24 ... (1)

dS/ds = dS/ds*ds/dt = 12s ds/dt = 12 ...(2)

Solve for s and ds/dt by substitution or dividing (1) by (2).

I get s=8, and ds/dt=1/8.

- Calculus -
**fellani**, Tuesday, March 24, 2015 at 11:02pm
8

- Calculus -
**fellani**, Tuesday, March 24, 2015 at 11:02pm
5

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